Source parameters for historical earthquakes worldwide are compiled to develop a series of empirical relationships among moment magnitude ( M ), surface rupture length, subsurface rupture length, downdip rupture width, rupture area, and maximum and average displacement per event. The resulting data base is a significant update of previous compilations and includes the additional source parameters of seismic moment, moment magnitude, subsurface rupture length, downdip rupture width, and average surface displacement. Each source parameter is classified as reliable or unreliable, based on our evaluation of the accuracy of individual values. Only the reliable source parameters are used in the final analyses. In comparing source parameters, we note the following trends: (1) Generally, the length of rupture at the surface is equal to 75% of the subsurface rupture length; however, the ratio of surface rupture length to subsurface rupture length increases with magnitude; (2) the average surface displacement per event is about one-half the maximum surface displacement per event; and (3) the average subsurface displacement on the fault plane is less than the maximum surface displacement but more than the average surface displacement. Thus, for most earthquakes in this data base, slip on the fault plane at seismogenic depths is manifested by similar displacements at the surface. Log-linear regressions between earthquake magnitude and surface rupture length, subsurface rupture length, and rupture area are especially well correlated, showing standard deviations of 0.25 to 0.35 magnitude units. Most relationships are not statistically different (at a 95% significance level) as a function of the style of faulting: thus, we consider the regressions for all slip types to be appropriate for most applications. Regressions between magnitude and displacement, magnitude and rupture width, and between displacement and rupture length are less well correlated and have larger standard deviation than regressions between magnitude and length or area. The large number of data points in most of these regressions and their statistical stability suggest that they are unlikely to change significantly in response to additional data. Separating the data according to extensional and compressional tectonic environments neither provides statistically different results nor improves the statistical significance of the regressions. Regressions for cases in which earthquake magnitude is either the independent or the dependent parameter can be used to estimate maximum earthquake magnitudes both for surface faults and for subsurface seismic sources such as blind faults, and to estimate the expected surface displacement along a fault for a given size earthquake.

New Empirical Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Experimental results in the published literature show that at low normal stress the shear stress required to slide one rock over another varies widely between experiments. This is because at low stress rock friction is strongly dependent on surface roughness. At high normal stress that effect is diminished and the friction is nearly independent of rock type. If the sliding surfaces are separated by gouge composed of Montmorillonite or vermiculite the friction can be very low.

Friction of Rocks

Direct shear experiments on ground surfaces of a granodiorite from Raymond, California, at normal stresses of ∼6 MPa demonstrate that competing time, displacement, and velocity effects control rock friction. It is proposed that the strength of the population of points of contacts between sliding surfaces determines frictional strength and that the population of contacts changes continuously with displacements. Previous experiments demonstrate that the strength of the contacts increases with the age of the contacts. The present experiments establish that a characteristic displacement, proportional to surface roughness, is required to change the population of contacts. Hence during slip the average age of the points of contact and therefore frictional strength decrease as slip velocity increases. Displacement weakening and consequently the potential for unstable slip occur whenever displacement reduces the average age of the contacts. In addition to this velocity dependency, which arises from displacement dependency and time dependency, the experiments also show a competing but transient increase in friction whenever slip velocity increases. Creep of the sliding surface at stresses below that for steady state slip is also observed. Constitutive relationships are developed that permit quantitative simulation of the friction versus displacement data as a function of surface roughness and for different time and velocity histories. Unstable slip in experiments is controlled by these constitutive effects and by the stiffness of the experimental system. It is argued that analogous properties control earthquake instability.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB084iB05p02161%4010.1002/%28ISSN%292169-9356.FAULT1

Modeling of rock friction: 1. Experimental results and constitutive equations

T his paper investigates the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation. General conditions for a bifurcation, corresponding to the localization of deformation into a planar band, are derived. Although the analysis is general and applications to other localization phenomena are noted, the constitutive relations which are examined in application of the criterion for localization are intended to model the behavior of brittle rock masses under compressive principal stresses. These relations are strongly pressure-sensitive since inelasticity results from frictional sliding on an array of fissures; the resulting inelastic response is dilatant, owing to uplift in sliding at asperities and to local tensile cracking from fissure tips. The appropriate constitutive descriptions involve non-normality of plastic strain increments to the yield hyper-surface. Also, it is argued that the subsequent yield surfaces will develop a vertex-like structure. Both of these features are shown to be destabilizing and to strongly influence the resulting predictions for localization by comparison to predictions based on classical plasticity idealizations, involving normality and smooth yield surfaces. These results seem widely applicable to discussions of the inception of rupture as a constitutive instability.

http://esag.harvard.edu/rice/057_RudnickiRi_localization_JMPS75.pdf

Conditions for the localization of deformation in pressure-sensitive dilatant materials

The traditional view of tectonics is that the lithosphere comprises a strong brittle layer overlying a weak ductile layer, which gives rise to two forms of deformation: brittle fracture, accompanied by earth- quakes, in the upper layer, and aseismic ductile flow in the layer beneath Although this view is not incorrect, it is imprecise, and in ways that can lead to serious misunderstandings The term ductility, for example, can apply equally to two common rock deformation mechanisms: crystal plasticity, which occurs in rock above a critical temperature, and cataclastic flow, a type of granular deformation which can occur in poorly consolidated sediments Although both exhibit ductility, these two deformation mechanisms have very different rheologies Earthquakes, in turn, are associated with strength and brittleness—associations that are likewise sufficiently imprecise that, if taken much beyond the generality implied in the opening sentence, they can lead to serious misinterpretations about earthquake mechanics Lately, a newer, much more precise and predictive model for the earthquake mechanism has emerged, which has its roots in the observation that tectonic earthquakes seldom if ever occur by the sudden appearance and propagation of a new shear crack (or 'fault') Instead, they occur by sudden slippage along a pre-existing fault or plate interface They are therefore a frictional, rather than fracture, phenomenon, with brittle fracture playing a secondary role in the lengthening of faults 1 and frictional wear 2  This distinction was noted by several early workers 3 , but it was not until 1966 that Brace and Byerlee 4 pointed out that earthquakes must be the result of a stick-slip frictional instability Thus, the earthquake is the 'slip', and the 'stick' is the interseismic period of elastic strain accumula- tion Subsequently, a complete constitutive law for rock friction has been developed based on laboratory studies A surprising result is that a great many other aspects of earthquake phenomena also now seem to result from the nature of the friction on faults The properties traditionally thought to control these processes— strength, brittleness and ductility—are subsumed within the over- arching concept of frictional stability regimes Constitutive law of rock friction In the standard model of stick-slip friction it is assumed that sliding begins when the ratio of shear to normal stress on the surface reaches a value ms, the static friction coefficient Once sliding initiates, frictional resistance falls to a lower dynamic friction coefficient, md, and this weakening of sliding resistance may,

/pdf/earthquakes-and-friction-laws-22mbgrykdx.pdf

Earthquakes and friction laws

Volume changes of a granite, a marble, and an aplite were measured during deformation in triaxial compression at confining pressure of as much as 8 kb. Stress-volumetric strain behavior is qualitatively the same for these rocks and a wide variety of other rocks and concrete studied elsewhere. Volume changes are purely elastic at low stress. As the maximum stress becomes one-third to two-thirds the fracture stress at a given pressure, the rocks become dilatant; that is, volume increases relative to elastic changes. The magnitude of the dilatancy, with a few exceptions, ranges from 0.2 to 2.0 times the elastic volume changes that would have occurred were the rock simply elastic. The magnitude of the dilatancy is not markedly affected by pressure, for the range of conditions studied here.

For granite, the stress at which dilatancy was first detected was strongly time dependent; the higher the loading rate the higher the stress. Dilatancy, which represents an increase in porosity, was traced in the granite to open cracks which form parallel with the direction of maximum compression.

Dilatancy in the Fracture of Crystalline Rocks

During the deformation of rock in laboratory experiments, small cracking events, i.e., microfractures, occur which radiate elastic waves in a manner similar to earthquakes. These radiations were detected during uniaxial and triaxial compression tests and their frequency-magnitude relation studied. They were found to obey the Gutenberg and Richter relation log N = a + b M 
Where N is the number of events which occurred of magnitude M , and a and b constants. The dependence of the parameter b on rock type, stress, and confining pressure was studied. It was found to depend primarily on stress, in a characteristic way. The frequency-magnitude relation for events which accompanied frictional sliding and deformation of a ductile rock was found to have a much higher b value than that observed in brittle rock. The Gutenberg and Richter formulation of the frequency-magnitude relation was derived from a statistical model of rock and crustal deformation. This analysis demonstrates the basis of similarity between rock deformation experiments in the laboratory and deformation of the crust.

The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes

http://science.sciencemag.org/content/sci/181/4102/803.full.pdf

Christopher H. Scholz

Papers

The Mechanics of Earthquakes and Faulting

Earthquakes and friction laws

Dilatancy in the Fracture of Crystalline Rocks

The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes

Earthquake Prediction: A Physical Basis